Above and below Guarantee Parameterizations for Combinatorial Optimisation Problems

Parameterized complexity can be viewed as a two-dimensional approach to traditional complexity theory. Rather than measuring the complexity of a problem purely in terms of the input size, we also consider some structural parameter of the input instance, and measure how the complexity of the problem changes as this parameter changes. The analogue of polynomial time tractability is fixed-parameter tractability, in which the problem can be solved efficiently as long as the value of the parameter remains small. It turns out that for many practical applications of traditionally NP-complete problems, the problem is fixed-parameter tractable for some choice of parameter which in practice is often small. Parameterized complexity thus provides a theoretically rigorous framework for explaining why so many applications of NPcomplete problems are in practice solvable efficiently. It is natural to ask how the complexity of a problem changes with the size of the solution sought. Aboveand below-guarantee parameterizations are a way to model this in parameterized complexity theory. Given an optimisation problem with a known lower (upper) bound on the value of the objective function, we ask how the complexity of the problem changes as the desired solution value gets further above (below) the bound. In the first part of this thesis, we consider above-bound parameterizations for a family of graph theory problems related to Max-Cut, in which we are given a graph and asked to find a maximum bipartite subgraph. The Edwards-Erdős bound gives a lower bound on the size of such a subgraph, and the parameterization of Max-Cut above this bound was an open problem for 15 years. We show that this problem is fixed-parameter extendible, and extend this result to the weighted case for a family of problems that generalizes Max-Cut. For the non-weighted case, we also prove polynomial kernel results (in which an instance can be reduced to an equivalent instance with size bounded by a polynomial in the parameter) for the corresponding parameterizations of Balanced Subgraph (a problem that generalizes Max-Cut) and Max Acyclic Subgraph on oriented graphs. In the second part of this thesis, we investigate various below-guarantee parameterizations for two hypergraph problems, Hitting Set and Test-Cover. We also use our results for the former problem to show positive results for other parameterized problems, including an above-guarantee parameterization of Max-SAT.